mohanty59supriya

2022-07-16

Prove that if x is a rational number and y is an irrational number,then x+y is an irrational number. If in addition, x≠0, then show that xy is an irrational number.

Jeffrey Jordon

Expert2022-11-07Added 2605 answers

Hint: Consider $(x+y)-x$

**Explanation:**

As is very often the case, we do not *need* to write this as a proof by contradiction. We can prove the contrapositive directly.

We can prove directly:

x is rational $\Rightarrow $ (x+y is rational $\Rightarrow $ y is rational)

(using $a,b\in Q\Rightarrow a-b\in Q$ -- that is, Q is closed under subtraction)

Therefore (by contraposition of the imbedded conditional)

x is rational $\Rightarrow $ (y is not rational $\Rightarrow $ x+y is not rational)

This is logically equivalent to

(x is rational & y is not rational) $\Rightarrow $ x+y is not rational)

**By contradiction**

Suppose p and $\neg q$ and r .

Prove a contradiction and conclude that if p and $\neg q$, then $\neg r$

**By contrapositive**

Suppose p and $\neg r$. Prove that q.

Conclude that If p and $\neg q$, then r.

The two methods are very closely related and I don't know of anyone who accepts one and not the other. (Although many/most/all intuitionists refuse to accept either contradiction or contrapositive.)

**Proof by contrapositive**

Suppose that x is rational and $x+y$ is rational.

Then the difference $(x+y)-x=y$ is rational.

Hence is we know that y is irrational, then $x+y$ must have been irrational.

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